Why is it better to promote your characters at 15 instead of 20?
Don't they get additional boosts from those 5 levels?
They do, but since there's no set of maximum levels, you don't really need to wait until the maximum level cap before promotion.
Plus, every time you re-class, levels are added to your cumulative level. The formula for this is:
current level -1[/2], rounded down
So if you re-classed at level 20, you would add +9 to your Cumulative Level [20 -1/2 = 9.5 = 9 rounded down]. If you re-classed at 15, you would add +7 [15 -1/2 = 7].
It's better to keep these levels low, since the higher they are the less XP you acquire from enemies. Keep in mind that when you Promote and then re-class, it increases the Cumulative Level by a nice bit [since Promoted Classes automatically have a Current Level of 20+]. That's why when you Promote someone and then re-class to a Base Class the XP gain seems so much slower.
I should also mention that there's a cap to the amount your Cumulative Level can reach. On Normal it's 20, Hard it's 30, and on Lunatic it's 50.
You'll never stop gaining XP permanently, it'll just decrease to a maximum of say...8 per enemy.